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Mirrors > Home > ILE Home > Th. List > eluniab | GIF version |
Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
eluniab | ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni 3739 | . 2 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})) | |
2 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦 | |
3 | nfsab1 2129 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
4 | 2, 3 | nfan 1544 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) |
5 | nfv 1508 | . . 3 ⊢ Ⅎ𝑦(𝐴 ∈ 𝑥 ∧ 𝜑) | |
6 | eleq2 2203 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)) | |
7 | eleq1 2202 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})) | |
8 | abid 2127 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
9 | 7, 8 | syl6bb 195 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) |
10 | 6, 9 | anbi12d 464 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝑥 ∧ 𝜑))) |
11 | 4, 5, 10 | cbvex 1729 | . 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑)) |
12 | 1, 11 | bitri 183 | 1 ⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1468 ∈ wcel 1480 {cab 2125 ∪ cuni 3736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-uni 3737 |
This theorem is referenced by: elunirab 3749 dfiun2g 3845 inuni 4080 snnex 4369 elfv 5419 unielxp 6072 tfrlem9 6216 tfr0dm 6219 metrest 12675 |
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